# Analysis Seminar

#### Remarks on Onsager's Conjecture and Anomalous Dissipation on domains with and without boundaries

Speaker: Theodore Drivas, Princeton University

Location: Warren Weaver Hall 1302

Date: Thursday, April 5, 2018, 11 a.m.

Synopsis:

We​ ​first discuss the inviscid limit of the global energy dissipation of
Leray solutions of incompressible​ ​Navier-Stokes on the torus.  Assuming
that the solutions have Besov norms bounded uniformly in viscosity, we
establish an upper​ ​bound on energy dissipation. As a consequence, Onsager-type​
"quasi-singularities" are required in the Leray solutions, even if the total
energy dissipation is o(ν) in the limit ν → 0.  Next, we​ ​discuss​ ​an extension
of Onsager's conjecture for domains with solid boundaries.​ ​We give a localized
regularity condition​ ​for energy conservation of weak solutions of the Euler
equations​​ assuming Besov regularity of the velocity with σ>1/3 for any U⋐Ω​
and, on an arbitrary thin layer around the boundary,​ ​boundedness of velocity,
pressure and continuity​ ​of the wall-normal velocity. We also prove that the global
viscous dissipation vanishes in the inviscid​ ​limit for Leray-Hopf solutions of the
Navier-Stokes equations under the similar assumptions, but​ ​holding uniformly in a​
vanishingly thin viscous boundary layer.  Finally, if a strong Euler solution exists,
we show that equicontinuity at the boundary within a O(ν) strip alone suffices to
conclude the absence of anomalous dissipation.

The first part of the talk concerns joint work​ ​with G. Eyink, the second with H.Q. Nguyen.